Note: This means that \(i^i\) is not a well defined (unique) quantity. To remedy this, one needs to specify a branch cut. For example, we can define the argument of \(e^{i \theta}\) to be defined for \(\theta \in [0, 2\pi)\), in which case we have that \(i^i = e^{-\pi / 2}\). That is, this forces \(k = 0\). Of course, different branch cut can be chosen yielding different values for \(k\).

Geometric Interpretation

Euler's formula allows for any complex number \( x \) to be represented as \( e^{ix} \), which sits on a unit circle with real and imaginary components \( \cos{x} \) and \( \sin{x} \), respectively. Various operations (such as finding the roots of unity) can then be viewed as rotations along the unit circle.

Trigonometric Applications

One immediate application of Euler's formula is to extend the definition of the trigonometric functions to allow for arguments that extend the range of the functions beyond what is allowed under the real numbers.

A couple useful results to have at hand are the facts that

\[ e^{-ix} = \cos{x} - i \sin{x}, \]

so

\[ e^{ix} + e^{-ix} = 2 \cos{x}. \]

It follows that

\[ \cos{x} = \frac{e^{ix} + e^{-ix}}{2}, \]

and similarly

\[ \sin{x} = \frac{e^{ix} - e^{-ix}}{2i} \]

and

\[ \tan{x} = \frac{e^{ix} - e^{-ix}}{i(e^{ix} + e^{-ix})}. \]

Solve \( \cos{x} = 2 \) in the complex numbers.

We first note that if \(x = x_0 \) is a solution, then so is \(x = 2\pi k \pm x_0 \) for any integer \(k\). This is because \(\cos x\) is an even function with a fundamental period of \(2\pi \).

De Moivre's theorem has many applications. As an example, one may wish to compute the roots of unity, or the complex solution set to the equation \( x^n = 1 \) for integer \( n \). Notice that \( e^{2\pi ki} \) is always equal to \( 1 \) for \( k \) an integer, so the \( n^\text{th} \) roots of unity must be